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Theorem xpdisj1 4931
Description: Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
Assertion
Ref Expression
xpdisj1  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  C )  i^i  ( B  X.  D ) )  =  (/) )

Proof of Theorem xpdisj1
StepHypRef Expression
1 inxp 4641 . 2  |-  ( ( A  X.  C )  i^i  ( B  X.  D ) )  =  ( ( A  i^i  B )  X.  ( C  i^i  D ) )
2 xpeq1 4521 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  B )  X.  ( C  i^i  D ) )  =  (
(/)  X.  ( C  i^i  D ) ) )
3 0xp 4587 . . 3  |-  ( (/)  X.  ( C  i^i  D
) )  =  (/)
42, 3syl6eq 2164 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  B )  X.  ( C  i^i  D ) )  =  (/) )
51, 4syl5eq 2160 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  C )  i^i  ( B  X.  D ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    i^i cin 3038   (/)c0 3331    X. cxp 4505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958  df-xp 4513  df-rel 4514
This theorem is referenced by:  djudisj  4934  xp01disjl  6297  xpfi  6784  djuinr  6914
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